On energy-critical half-wave maps into \({\mathbb {S}}^2\)

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Abstract

We consider the energy-critical half-wave maps equation
$$\begin{aligned} \partial _t {\mathbf {u}}+ {\mathbf {u}}\wedge |\nabla | {\mathbf {u}}= 0 \end{aligned}$$
for \({\mathbf {u}}: [0,T) \times {\mathbb {R}}\rightarrow {\mathbb {S}}^2\). We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on \({\mathbb {S}}^2\). In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Möbius group on the target \({\mathbb {S}}^2\) applied to half-harmonic maps from \({\mathbb {R}}\) to \({\mathbb {S}}^2\). Complementing our classification result, we carry out a detailed analysis of the linearized operator L around half-harmonic maps \({\mathbf {Q}}\) with arbitrary degree \(m \geqslant 1\) and consisting of m identical Blaschke factors. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of \({\mathbf {Q}}\). Moreover, we give a full description of the spectrum of L by finding all its \(L^2\)-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for L and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on \({\mathbb {R}}\) to the unit circle \({\mathbb {S}}\). Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.

Notes

Acknowledgements

It is a pleasure to thank G. M. Graf and R. L. Frank for helpful conversations regarding the action of the Lorentz group on \({\mathbb {S}}^2\) and the use of the stereographic projection in the spectral analysis, respectively. E. Lenzmann acknowledges financial support by the Swiss National Science Foundation (SNF) and he is also grateful for the kind hospitality of the I.H.É.S., where part of this work was done. A. Schikorra is supported by the German Research Foundation (DFG) through Grant No. SCHI-1257-3-1 and a Heisenberg fellowship. Finally, the authors would like to thank one anonymous referee for her/his very careful reading of this work and helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA

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